Laplace equation in polars

Laplace's equation in 2 dimensions may be written, using plane polar coordinates r, θ, as

Find all separable solutions of this equation which have the form V(r, θ)=R(r)S(θ), which are single valued for all r, θ. What property of the equation makes any linear combination of such solutions also a solution?

2. Relevant equations

3. The attempt at a solution
i get how to separate the variables and i am left with to expressions equaling a constant. but from there it doesnt make sense to me how to end up with 2 solutions

assume a seperable solution then sub it in and do the derivatives. then multiply thorugh by [itex]\frac{r^2}{RS}[/itex]

if you make your constant [itex]\lambda[/itex] you should get a nice solution for [itex]\lambda>0[/itex] hint : let [itex]R=r^m[/itex] with m a constant. the others will be less pretty unless you have simplifying boundary conditions.

ok, got it thanks. the second part of the question which i didnt include before is:
A continuous potential V(r, θ) satisfies Laplace's equation everywhere except on the concentric circles r=a, r=b where b>a.
(i) Given that V(r=a, θ)=Vo(1+cos θ), and that V is finite as r-->infinity , find V in the region r less than or equal to a
(ii) given, separately, that V(r=0, θ)=2Vo and V is finite as r--> ∞, find V for r≥b

for (i), i dont really know how to select solutions for that V. my solutions arecombinations of sinmtheta cosmtheta r^m and r^-m. is there a way to do this through an expansion?

ok i think that the periodicity of the cos term means that the solutions for [itex]\lambda<0[/itex] are useless here and the [itex]\lambda=0[/itex] solutions will be trivial after you apply the b.c. that it must be finit as r goes to infinity.

so for the [itex]\lambda>0[/itex] solutions i have

[itex]V=(Ar^n+Br^{-n})(C \cos{n \theta} + D \sin{n \theta})[/itex]

for this to be finite at infinity what can you say about the coefficient of [itex]r^n[/itex]